Inverse optimal robust control of singularly impulsive dynamical systems

In this paper for the class of nonlinear uncertain singularly impulsive dynamical systems we present optimal robust control and inverse robust optimal control results. We consider a control problem for nonlinear uncertain singularly impulsive dynamical systems involving a notion of optimality with respect to an auxiliary cost which guarantees a bound on the worst-case value of a nonlinear-nonquadratic hybrid cost criterion over a prescribed uncertainty set. Further we specialize result to affine uncertain systems to obtain controllers predicated on an inverse optimal hybrid control problem. In particular, to avoid the complexity in solving the steady-state hybrid Hamilton-Jacobi-Bellman equation we parameterize a family of stabilizing hybrid controllers that minimize some derived hybrid cost functional that provides flexibility in specifying the control law. The performance integrand is shown to explicitly depend on the nonlinear singularly impulsive system dynamics, the Lyapunov function of the closed-loop system, and the stabilizing hybrid feedback control law wherein the coupling is introduced via the hybrid Hamilton-Jacobi-Bellman equation. By varying the parameters in the Lyapunov function and the performance integrand, the proposed framework can be used to characterize a class of globally stabilizing hybrid controllers that can meet the closed-loop system response constraints. Obtained results for nonlinear case are further specialized to linear singularly impulsive dynamical systems with polynomial and multilinear performance functional.